A242207 -id:A242207 - cp's OEIS frontend

A242174 Least prime divisor of A005260(n) which does not divide any previous term A005260(k) with k < n, or 1 if such a primitive prime divisor of A005260(n) does not exist.

2, 3, 41, 5, 7, 349, 61, 75617, 31, 13, 499, 643897693, 17, 19, 1729774061, 101, 2859112064587, 138407, 83, 167, 59, 29, 653, 257, 997540809461453561581, 347, 13679, 37, 160449179727717672892660463, 211, 151, 43, 97, 73, 47

Offset: 1

Zhi-Wei Sun, May 07 2014

nonn

Conjecture: a(n) is prime for any n > 0. In general, for any r > 2, if n is large enough then f_r(n) = sum_{k=0..n}C(n,k)^r has a prime divisor which does not divide any previous terms f_r(k) with k < n.

			a(3) = 41 since A005260(3) = 2^2*41 with 41 dividing none of A005260(1) = 2 and A005260(2) = 2*3^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..82

Cf. A000040, A005260, A242169, A242170, A242171, A242173, A242193, A242194, A242195, A242207.

u[n_]:=Sum[Binomial[n,k]^4,{k,0,n}]
f[n_]:=FactorInteger[u[n]]
p[n_]:=Table[Part[Part[f[n],k],1],{k,1,Length[f[n]]}]
Do[If[u[n]<2,Goto[cc]];Do[Do[If[Mod[u[i],Part[p[n],k]]==0,Goto[aa]],{i,1,n-1}];Print[n," ",Part[p[n],k]];Goto[bb];Label[aa];Continue,{k,1,Length[p[n]]}];Label[cc];Print[n," ",1];Label[bb];Continue,{n,1,35}]

A242223 Least prime p such that H(n) == 0 (mod p) but H(k) == 0 (mod p) for no 0 < k < n, or 1 if such a prime p does not exist, where H(n) denotes the n-th harmonic number sum_{k=1..n}1/k.

Original entry on oeis.org

1, 3, 11, 5, 137, 7, 1, 761, 7129, 61, 97, 13, 29, 1049, 41233, 17, 37, 19, 7440427, 11167027, 18858053, 23, 583859, 577, 109, 34395742267, 521, 375035183, 4990290163, 31, 2667653736673, 2917, 269, 3583, 397, 1297, 10839223, 199, 737281, 41

Offset: 1

Zhi-Wei Sun, May 08 2014

nonn

Conjecture: a(n) is prime except for n = 1, 7.

			a(4) = 5 since H(4) = 25/12 == 0 (mod 5), but none of H(1) = 1, H(2) = 3/2 and H(3) = 11/6 is congruent to 0 modulo 5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..184
Z.-W. Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014.

Cf. A000040, A001008, A002805, A242169, A242170, A242171, A242173, A242174, A242193, A242194, A242195, A242207, A242222.

h[n_]:=Numerator[HarmonicNumber[n]]
f[n_]:=FactorInteger[h[n]]
p[n_]:=Table[Part[Part[f[n],k],1],{k,1,Length[f[n]]}]
Do[If[h[n]<2,Goto[cc]];Do[Do[If[Mod[h[i],Part[p[n],k]]==0,Goto[aa]],{i,1,n-1}];Print[n," ",Part[p[n],k]];Goto[bb];Label[aa];Continue,{k,1,Length[p[n]]}];Label[cc];Print[n," ",1];Label[bb];Continue,{n,1,40}]

A242292 Least prime divisor of 2^n - n which does not divide any 2^k - k with 0 < k < n, or 1 if such a primitive prime divisor of 2^n - n does not exist.

Original entry on oeis.org

1, 2, 5, 3, 1, 29, 11, 31, 503, 13, 7, 1021, 8179, 1637, 4679, 1, 8737, 131063, 524269, 262139, 2097131, 349, 131, 773, 271, 197, 457, 1493, 317, 17, 6733, 73, 41, 157109, 83, 53, 1741, 3329, 49977801259, 997, 149, 2199023255531, 61, 4398046511093, 3769453

Offset: 1

Zhi-Wei Sun, May 10 2014

nonn

Conjecture: a(n) = 1 only for n = 1, 5, 16.

In constrast, a classical theorem of Bang asserts that if n > 1 is different from 6 then 2^n - 1 has a prime divisor which does not divide any 2^k - 1 with 0 < k < n.

			a(4) = 3 since 2^4 - 4 = 2^2*3 with 3 dividing none of 2^1 - 1 = 1, 2^2 - 2 = 2 and 2^3 - 3 = 5.

A. S. Bang, Taltheoretiske Undersgelser, Tidsskrift fur Mat. 4(1886), no. 5, 70--80, 130--137.

Zhi-Wei Sun, Table of n, a(n) for n = 1..215
Z.-W. Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014.

Cf. A000040, A000325, A112927, A242169, A242170, A242171, A242173, A242174, A242193, A242194, A242195, A242207.

u[n_]:=2^n-n
f[n_]:=FactorInteger[u[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[u[n]<2, Goto[cc]]; Do[Do[If[Mod[u[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 45}]

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A242174 Least prime divisor of A005260(n) which does not divide any previous term A005260(k) with k < n, or 1 if such a primitive prime divisor of A005260(n) does not exist.

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A242223 Least prime p such that H(n) == 0 (mod p) but H(k) == 0 (mod p) for no 0 < k < n, or 1 if such a prime p does not exist, where H(n) denotes the n-th harmonic number sum_{k=1..n}1/k.

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A242292 Least prime divisor of 2^n - n which does not divide any 2^k - k with 0 < k < n, or 1 if such a primitive prime divisor of 2^n - n does not exist.

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